Internal geometry and functors between sites

TL;DR

This paper investigates how different Grothendieck topologies relate and how functors between categories preserve locality, with applications to geometric objects like sheaves and groupoids across various sites.

Contribution

It introduces definitions ensuring invariance of geometric objects under topology equivalences and functors, broadening understanding of locality in category theory.

Findings

Invariance of sheaves, groupoids, and functors under topology changes.

Relations between Grothendieck topologies on a single category.

Analysis of functors between categories of manifolds, spaces, and sheaves.

Abstract

Locality is implemented in an arbitrary category using Grothendieck topologies. We explore how different Grothendieck topologies on one category can be related, and, more general, how functors between categories can preserve them. As applications of locality, we review geometric objects such as sheaves, groupoids, functors, bibundles, and anafunctors internal to an arbitrary Grothendieck site. We give definitions such that all these objects are invariant under equivalences of Grothendieck topologies and certain functors between sites. As examples of sites, we look at categories of smooth manifolds, diffeological spaces, topological spaces, and sheaves, and we study properties of various functors between those.